On Arnold-type stability theorems for the Euler equation on a sphere

In this paper, we establish three Arnold-type stability theorems for steady or rotating solutions of the incompressible Euler equation on a sphere.

Specifically, we prove that if the stream function of a flow solves a semilinear elliptic equation with a monotone nonlinearity, then, under appropriate conditions, the flow is stable or orbitally stable in the Lyapunov sense.

In particular, our theorems apply to degree-2 Rossby-Haurwitz waves.

These results are achieved via a variational approach, with the key ingredient being to show that the flows under consideration satisfy the conditions of two Burton-type stability criteria which are established in this paper.

As byproducts, we obtain some sharp rigidity results for solutions of semilinear elliptic equations on a sphere.